Team Selection Test 2009 #0ivw

Team Selection Test 2009Hard

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Understand the scale

Scale 0.5: Easiest math competition problems, often solvable even for early middle school students without experience. 1: Problems strictly for beginners, often at a moderate middle school level. 1.5: Problems for stronger beginner students, on the level of the middling problems in most middle school contests. 2: For motivated beginners, harder questions from the previous categories. 2.5: More advanced beginner problems, hardest questions from previous categories. 3: Early intermediate problems that require more creative thinking. 3.5: Tougher early intermediate problems that consistently stretch into higher-level creative thinking or conceptual knowledge. 4: Intermediate-level problems. 4.5: Upper intermediate problems approaching the upper bound for non-invitational sprint math competitions. 5: More difficult AIME problems or simple proof-based Olympiad-style problems. 6: High-level AIME-styled questions or introductory Olympiad-level questions. 7: Tougher Olympiad-level questions, may require more technical knowledge. 8: High-level Olympiad-level questions. 9: Expert Olympiad-level questions. 9.5: The hardest problems appearing on Olympiads which the strongest students could reasonably solve. 10: Historically notorious problems, generally unsuitable for even very hard competitions because they are exceedingly tedious, long, and difficult.

Polynomial interpolation: Newton, LagrangeAlgebraic properties of binomial coefficientsGreatest common divisors (gcd)Least common multiples (lcm)

Problem

For each positive integer $n$ , let $c(n)$ be the largest real number such that

c(n) \leq \left| \frac{f(a) - f(b)}{a - b} \right|

for all triples $(f, a, b)$ such that

$f$ is a polynomial of degree $n$ taking integers to integers, and
$a, b$ are integers with $f(a) \neq f(b)$ .

Find $c(n)$ .

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